arXiv subjects

Alexander Dunn

Publications and source records attributed to Alexander Dunn.

20 records · Page 2Linked to original sources

arXiv2018

Polynomial partition asymptotics

Let $f \in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N}) \subseteq \mathbb{N}$, and let $p_{\mathcal{A}_{f}}(n)$ denote number of partitions of $n$ whose parts lie in the set $\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}$. Under hypotheses on the roots of $f-f(0)$, we use the Hardy--Littlewood circle method, a polylogarithm identity, and the Matsumoto--Weng zeta function to derive asymptotic formulae for $p_{\mathcal{A}_f}(n)$ as $n$ tends to infinity. This generalises asymptotic formulae for the number of partitions into perfect $d$th powers, established by Vaughan for $d=2$, and Gafni for the case $d \geq 2$, in 2015 and 2016 respectively.

math.NT↗

arXiv2017

On the distribution of $αp^γ+β$ modulo one

Let $\|\cdot\|$ denote the minimum distance to an integer. For $0<γ< 1$, $θ>0$ and $(α, β) \in \mathbb{R} \setminus \{0\} \times \mathbb{R}$ we study when \begin{equation*} \|αp^γ+β\|<p^{-θ}, \end{equation*} holds for infinitely many primes $p$ of a special type. In particular, we consider when this inequality holds for primes $p$ such that $p+2$ has few prime factors counted with multiplicity. This is done using an exponential sum estimate of the author and the linear sieve of Iwaniec with bilinear error term. This is related to recent work of Tolev, Todorova, Matömaki and Cai.

math.NT↗